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Dimension of a Set of Transformations

  1. Apr 28, 2012 #1
    If we consider the set R of all linear transformations from an p-dimensional vector space Z to Z (T:Z -> Z), what do we know about the dimension of the set R?

    In other words, what do we know about any basis for R? What are its properties?
     
  2. jcsd
  3. Apr 28, 2012 #2

    micromass

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    Hint: those linear mappings can be represented by matrices. What's the dimension of this space of matrices?
     
  4. Apr 28, 2012 #3
    Well, we know that each matrix is pxp, because Z is p-dimensional. The dimension of this space of matrices is p-squared (for instance there are 4 basis matrices for the 2x2 case).
    But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?
     
  5. Apr 28, 2012 #4

    micromass

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    Huh, what do you mean?? Every matrix is linear.
     
  6. Apr 28, 2012 #5
    Thanks for staying with me!


    I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

    On another note, is there a linear transformation that maps every vector in a vector space to zero?
     
  7. Apr 28, 2012 #6

    micromass

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    It does represent a linear function, namely f(x,y)=(y,0). Every matrix represents a linear function, that's why matrices are so important.

    Yep, that is a linear transformation. It's the analog of the zero matrix.
     
  8. Apr 28, 2012 #7
    Thanks for your help!

    Just to make sure, the dimension is p-squared, right?
     
  9. Apr 28, 2012 #8

    micromass

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    Right!
     
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